Nonuniqueness in spin-density-functional theory on lattices

In electronic many-particle systems, the mapping between densities and spin magnetizations, $\{n\left(\mathbf{r}\right),\mathbf{m}\left(\mathbf{r}\right)\}$, and potentials and magnetic fields, $\{v\left(\mathbf{r}\right),\mathbf{B}\left(\mathbf{r}\right)\}$, is known to be nonunique, which has fundamental and practical implications for spin-density-functional theory (SDFT). This paper studies the nonuniqueness (NU) in SDFT on arbitrary lattices. Two new, nontrivial cases are discovered, here called local saturation and global noncollinear NU, and their properties are discussed and illustrated. In the continuum limit, only some well-known special cases of NU survive.

Figure 1

Global noncollinear NU for a linear three-point lattice with $m_1=m_2+m_3$. All magnetic fields shown here, where ${\mathbf{B}}_1^{\prime }={\mathbf{B}}_1+\lambda {\mathbf{m}}_1∕m_1$ and ${\mathbf{B}}_{2,3}^{\prime }={\mathbf{B}}_{2,3}-\lambda {\mathbf{m}}_{2,3}∕m_{2,3}$ (see Table ), produce the same four-density (keeping $v$ fixed).